Beetlebrot

[Schlega]

Hi all,

   I was curious to see what happens if you replace e^2ln(z)+c with e^(ln(z))2. Here's the overall shape:



The crest on it's head:




The tip of it's abdomen:


The mouth:


<a href="http://www.youtube.com/v/5I2KhmT_Ybo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/5I2KhmT_Ybo&rel=1&fs=1&hd=1</a>

[kram1032]

yet another great variation of the default formulae

[Timeroot]

That's one of the coolest Mandelbrot variations I've ever seen (that still retain its form, to some extent). Many others involve things like the Mandelbrot with big chunks attached at every Misiurwecz point, or just being a distorted version of a multibrot, etc...

This one is very original. I would love to see a parameter shift of e^(ln(x)*(2*n+ln(x)*(1-n))) from n=1 (the default Mandelbrot) to n=0 (this one). I think parameter animations always provide a new layer of insight into how a fractal "gets" its shape. 

[Schlega]

Thanks, guys. I'm uploading that interpolation right now, but I did it in 1080p and I have a slow connection. In about 10 hours, it'll be up.

Also on the to do list:
-Make an animation using z^{2n(ln(z))(1-n)}
-Explore z^a(ln(z))b+c
-Use the triplex formulas for exp(z) and ln(z) to make a beetlebulb.

[ker2x]

I like it ! *hugs* 

[Schlega]

*hugs ker2x*

Youtube didn't like the uncompressed HD version, so I had to reduce the quality, but it's up now.

<a href="http://www.youtube.com/v/W7dFONTB0kk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/W7dFONTB0kk&rel=1&fs=1&hd=1</a>

[Timeroot]

Youtube didn't like the uncompressed HD version, so I had to reduce the quality, but it's up now.

<a href="http://www.youtube.com/v/W7dFONTB0kk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/W7dFONTB0kk&rel=1&fs=1&hd=1</a>

Tantalizing! What I wouldn't give for a computer I could explore that on.... 

[Schlega]

Here's an alternate interpolation:

<a href="http://www.youtube.com/v/7mwxluQo6iQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/7mwxluQo6iQ&rel=1&fs=1&hd=1</a>

[Schlega]

This is what the triplex definitions of exp and log do:


The spew would probably be reduced with higher iterations, but my computer isn't fast enough to explore it in detail.

Here's an exponentialish version:

expish(x,y,z) = exp(x)*(cos(y)cos(z),sin(y)*cos(z), sin(z)),
logish(x,y,z) = (ln(x²+y²+z²),atan2(x+iy),asin(z/r))
<a href="http://www.youtube.com/v/Yip3BDZF_Wc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Yip3BDZF_Wc&rel=1&fs=1&hd=1</a>

[kram1032]

doesn't look that buggy anymore, now.
But still very nice

Which definition of exp and log did you use in the other image?
Whoa, it looks like the bug didn't survive that xD

[Timeroot]

I'm guessing he used the Maclaurin series for exponentiation/logarithms... you know, 1+z+z^2 / 2 + z^3 / 6 + z^4 / 24....

[Schlega]

I used the analytic formulas that were derived in the triplex algebra thread, which came from the series definition.

[kram1032]

ah, that one
A very complex formula.
Must have taken forever to even do such a low-iter version.

[Schlega]

Yeah, it took about half an hour for that picture. And now I have to do it again because I found a bug in the code. 

Here's what the video should have looked like:

<a href="http://www.youtube.com/v/2yjQnwBgi7s&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/2yjQnwBgi7s&rel=1&fs=1&hd=1</a>

[Schlega]

The long formulas looked even less interesting after I fixed the bug. Have some Julia sets instead.
<a href="http://www.youtube.com/v/NI5-FOksWGs&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/NI5-FOksWGs&rel=1&fs=1&hd=1</a>